Question: Let V be an n-dimensional vector space over F, and let f, g : V V be linear transformations. (a) Prove that if {v1, .

Let V be an n-dimensional vector space over F, and let f, g : V V be linear transformations.

(a) Prove that if {v1, . . . , vn} V forms a basis of eigenvectors for both f and g, then the transformations commute, i.e., f g = g f.

(b) Assume f g = g f, and suppose that f has n distinct eigenvalues over F. Prove that there exists a basis {v1, . . . , vn} of V that consists of vectors that are eigenvectors for both f and g. (Hint: Choose a basis of eigenvectors for f (why is that possible?). For each v in this basis, what can you say about f(g(v)), and then what does that tell you about g(v)?)

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